Trees. Make Money Fast! Bank Robbery. Ponzi Scheme. Stock Fraud Goodrich, Tamassia, Goldwasser Trees
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1 Part 4: Trees
2 Trees Make Money Fast! Stock Fraud Ponzi Scheme Bank Robbery 2013 Goodrich, Tamassia, Goldwasser Trees 2
3 Example: Family Tree 2013 Goodrich, Tamassia, Goldwasser Trees 3
4 Example: Unix File System 2013 Goodrich, Tamassia, Goldwasser Trees 4
5 What is a Tree In computer science, a tree is an abstract model of a hierarchical structure A tree consists of nodes with a parent-child relation Applications: Organization charts File systems Programming environments US Sales International Computers R Us Manufacturing Laptops Europe Asia Canada Desktops R&D 2013 Goodrich, Tamassia, Goldwasser Trees 5
6 What is a Tree (Daniel Geva) Root n1 Edge A node( all circles) Parent of n6 and n7 n2 left subtree n2 n3 Left son of n3 Right son of n3 n4 n5 n6 n7 Leaf (all green nodes) n8 n9 n10 n11 Node height number of edges on the longest path to a leaf Tree height height of the root Balanced Tree All non- leaf have two sons 2013 Goodrich, Tamassia, Goldwasser Trees 6
7 Tree Terminology Root node without parent (A) Internal node node with at least one child (A, B, C, F) Leaf (External node) node without children (E, I, J, K, G, H, D) Ancestors of a node: parent, grandparent, grand-grandparent, etc. Depth of a node: number of ancestors Height of a node: 1 + Max height of children (leaf height = 0) Height of a tree maximum depth of any node (3) Descendant of a node child, grandchild, grand-grandchild, etc. E Subtree: tree consisting of a node and its descendants B F I J K A G C H subtree D 2013 Goodrich, Tamassia, Goldwasser Trees 7
8 Tree ADT We use positions to abstract nodes, left key is return type: Generic methods: Integer len() Boolean is_empty() Iterator positions() Iterator iter() Accessor methods: position root() position parent(p) Iterator children(p) Integer num_children(p) Query methods: Boolean is_leaf(p) Boolean is_root(p) Update method: element replace (p, o) Additional update methods may be defined by data structures implementing the Tree ADT Note: A tree position is like a list index 2013 Goodrich, Tamassia, Goldwasser Trees 8
9 Abstract Tree Class in Python 2013 Goodrich, Tamassia, Goldwasser Trees 9
10 Preorder Traversal A traversal visits the nodes of a tree in a systematic manner In a preorder traversal, a node is visited before its descendants Application: print a structured document Algorithm preorder(v) visit(v) for each child w of v preorder (w) 1 Make Money Fast! 2 1. Motivations 2. Methods References Greed 1.2 Avidity Stock Fraud 2.2 Ponzi Scheme 2.3 Bank Robbery 2013 Goodrich, Tamassia, Goldwasser Trees 10
11 Postorder Traversal In a postorder traversal, a node is visited after its descendants Application: compute space used by files in a directory and its subdirectories 9 1 h1c.doc 3K 3 homeworks/ h1nc.doc 2K DDR.java 10K cs16/ Algorithm postorder(v) for each child w of v postorder (w) visit(v) programs/ Stocks.java 25K Robot.java 20K 8 todo.txt 1K 2013 Goodrich, Tamassia, Goldwasser Trees 11
12 Binary Trees A binary tree is a tree with the following properties: Each internal node has at most two children (exactly two for proper binary trees) The children of a node are an ordered pair We call the children of an internal node left child and right child Proper Binary Tree: every node is a leaf or must have exactly two children D Applications: arithmetic expressions decision processes searching B E A F C G LINK TO PYTHON CODE H I 2013 Goodrich, Tamassia, Goldwasser Trees 12
13 Arithmetic Expression Tree Binary tree associated with an arithmetic expression internal nodes: operators external nodes: operands Example: arithmetic expression tree for the expression (2 (a - 1) + (3 b)) a 1 3 b LINK TO PYTHON CODE 2013 Goodrich, Tamassia, Goldwasser Trees 13
14 Decision Tree Binary tree associated with a decision process internal nodes: questions with yes/no answer external nodes: decisions Example: dining decision Want a fast meal? Yes How about coffee? No On expense account? Yes No Yes No Starbucks Spike s Al Forno Café Paragon 2013 Goodrich, Tamassia, Goldwasser Trees 14
15 Properties of Proper Binary Trees Notation n number of nodes e number of external nodes i number of internal nodes h height Properties: e = i + 1 n = 2e - 1 h i h (n - 1)/2 e 2 h h log 2 e h log 2 (n + 1) Goodrich, Tamassia, Goldwasser Trees 15
16 BinaryTree ADT The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT Additional methods: position left(p) position right(p) position sibling(p) Update methods may be defined by data structures implementing the BinaryTree ADT LINK TO PYTHON CODE 2013 Goodrich, Tamassia, Goldwasser Trees 16
17 Inorder Traversal In an inorder traversal a node is visited after its left subtree and before its right subtree Application: draw a binary tree x(v) = inorder rank of v y(v) = depth of v 6 2 Algorithm inorder(v) if v has a left child inorder (left (v)) visit(v) if v has a right child inorder (right (v)) Goodrich, Tamassia, Goldwasser Trees 17
18 Print Arithmetic Expressions Specialization of an inorder traversal print operand or operator when visiting node print ( before traversing left subtree print ) after traversing right subtree + Algorithm printexpression(v) if v has a left child print( ( ) inorder (left(v)) print(v.element ()) if v has a right child inorder (right(v)) print ( ) ) 2-3 b a 1 ((2 (a - 1)) + (3 b)) LINK TO PYTHON CODE 2013 Goodrich, Tamassia, Goldwasser Trees 18
19 Evaluate Arithmetic Expressions Specialization of a postorder traversal recursive method returning the value of a subtree when visiting an internal node, combine the values of the subtrees + Algorithm evalexpr(v) if is_leaf (v) return v.element () else x evalexpr(left (v)) y evalexpr(right (v)) operator stored at v return x y LINK TO PYTHON CODE 2013 Goodrich, Tamassia, Goldwasser Trees 19
20 Euler Tour Traversal Generic traversal of a binary tree Includes a special cases the preorder, postorder and inorder traversals Walk around the tree and visit each node three times: on the left (preorder) from below (inorder) on the right (postorder) + L R 2 B Goodrich, Tamassia, Goldwasser Trees 20
21 Linked Structure for Trees A node is represented by an object storing Element Parent node Sequence of children nodes Node objects implement the Position ADT B B A D F A D F C E C E 2013 Goodrich, Tamassia, Goldwasser Trees 21
22 The Node Class class Node: "Class for storing a binary tree node" def init (self, element, parent=none, left=none, right=none): self.element = element self.parent = parent self.left = left self.right = right 2013 Goodrich, Tamassia, Goldwasser Trees 22
23 Linked Structure for Binary Trees A node is represented by an object storing Element Parent node Left child node Right child node B Node objects implement the Position ADT B A D A D C E C E 2013 Goodrich, Tamassia, Goldwasser Trees 23
24 Array-Based Representation of Binary Trees Nodes are stored in an array A 1 A 0 A B D G H B D Node v is stored at A[rank(v)] rank(root) = 1 if node is the left child of parent(node), rank(node) = 2 rank(parent(node)) if node is the right child of parent(node), rank(node) = 2 rank(parent(node)) E G F H 6 7 C J 2013 Goodrich, Tamassia, Goldwasser Trees 24
25 Example: Directory Disk Space 2013 Goodrich, Tamassia, Goldwasser Trees 25
26 Example: Directory Disk Space import os def disk_space(dir): size = 0 for file in os.listdir(dir) : path = dir + "/" + file if os.path.isfile(path): size += os.path.getsize(path) else: size += disk_space(path) return size 2013 Goodrich, Tamassia, Goldwasser Trees 26
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